To make the filling: In a small saucepan, whisk together the chicken broth and gelatin and let sit to bloom, about 5 minutes. Heat the mixture over low heat until melted, 2 to 3 minutes.
In a large cast-iron skillet, heat the olive oil over medium heat. Add the ground chicken and cook, breaking up the chicken well with a wooden spoon, until no longer pink, about 3 minutes. Add the garlic and cook, stirring constantly, until fragrant, about 1 minute. Stir in the soy sauce, mirin, sambal oelek, sesame oil, ginger, and a pinch of salt. Sauté, stirring frequently to break up the chicken into small pieces, until cooked through, about 1 minute. Turn off the heat and add the green onions.
On a small or quarter baking sheet, spread the ground chicken mixture into a single layer, then pour the gelatin mixture on top. Refrigerate until completely cool and the gelatin is set, about 20 minutes.
In a large bowl, add the set chicken and gelatin mixture and break it up with a wooden spoon.
Brush the perimeter of a dumpling wrapper with the beaten egg. Scoop a heaping tablespoon of the filling into the center of the wrapper and fold to seal into a half moon. Brush one point of the half moon with the beaten egg and join with the other point. Pinch to seal and place on a baking sheet. Repeat with the remaining wrappers, beaten egg, and filling.
Fill a large braiser or skillet halfway with water and bring to a boil over medium-high heat. Set the bottom of a three-basket bamboo steamer in the water. Line the other two levels of the steamer with parchment paper, spray with cooking spray, and poke some holes in the parchment. Add the dumplings to the lined steamer baskets, place on top, cover, and let steam, about 10 minutes.
To make Jo’s aunt’s special soy sauce: In a medium bowl, whisk together the soy sauce, water, gochugaru, sesame oil, white vinegar, sugar, and green onions.
Serve the dumplings with Jo’s aunt’s special soy sauce.